I found the solution. Maybe this was supposed to be obvious (it's obvious to me now!), but for anyone else who might run into this, here's what I discovered.

The LazyApproval Settings can be updated at the Farm, Site Collection, or Site level.

Changes at a lower level override the phrases at a higher level. (Site overrides site collection; site collection overrides farm.)

At some point in the past, the LazyApproval phrase "further review is required" was added to my site collection settings. This is where the workflow was running. Adding a phrase here breaks the inheritance, in a sense, and the set of phrases become independent from the farm's set of phrases.

Recently, I was adding LazyApproval phrases in Central Admin which is for the whole farm. Those weren't being recognized by the workflow in the site collection since the site collection phrases were overridden.

I stumbled across the list at the site collection level and realized it didn't have my new phrases I was trying to use. There is an option to "Revert back to the default LazyApproval term list for this server farm." (A similar option exists down at the subsite level.) I clicked this link and all the phrases that I had entered at the farm level showed up. I was then able to use those phrases in the the lazy approval email responses.